Finite Group DatabasebetaFinite Group Databasebeta| order | 24389 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C292 · Schur multiplier (H2) = C1 |
| extraspecial | 29−1+2 |
| unambiguous semidirect product | C841⋊C29 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C29 | C292 |
| prime radical | C292 | C29 |
| LCS | G ▷ C29 ▷ C1 | |
| UCS | C1 ◁ C29 ◁ G | |
| LECS | G ▷ C29 ▷ C1 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 841 |
| transitive degree | μt(G) = 841 |
A minimal faithful permutation representation — 2 generators on 841 points:
(1,2,4,8,15,26,42,64,93,130,176,228,269,306,341,376,411,442,473,504,535,566,597,628,659,690,721,752,783,3,6,10,17,28,44,66,95,132,178,230,271,308,343,378,413,444,475,506,537,568,599,630,661,692,723,754,785,814,7,13,20,31,47,69,98,135,181,233,274,311,346,381,415,446,477,508,539,570,601,632,663,694,725,756,787,784,815,14,24,36,52,74,103,140,186,238,278,315,350,385,380,417,449,481,513,545,577,609,641,673,705,737,769,753,786,816,25,40,58,82,111,148,194,245,284,282,322,360,398,383,419,451,483,515,547,579,611,643,675,707,739,722,755,788,817,41,62,87,119,159,205,193,252,237,283,323,361,345,384,420,452,484,516,548,580,612,644,676,708,691,724,757,789,818,63,91,124,165,212,209,198,256,241,287,326,364,348,387,422,454,486,518,550,582,614,646,678,660,693,726,758,790,819,92,128,170,217,214,210,201,258,243,289,328,310,349,388,423,455,487,519,551,583,615,647,629,662,695,727,759,791,820,129,174,222,216,215,211,203,185,244,290,329,313,352,390,425,457,489,521,553,585,617,598,631,664,696,728,760,792,821,175,226,263,219,158,147,204,189,248,293,273,314,353,391,426,458,490,522,554,586,567,600,633,665,697,729,761,793,822,227,267,301,220,161,152,208,191,250,295,276,317,355,393,428,460,492,524,556,536,569,602,634,666,698,730,762,794,823,268,304,262,164,162,155,139,192,251,232,277,318,356,394,429,461,493,525,505,538,571,603,635,667,699,731,763,795,824,305,339,265,167,163,156,143,197,255,235,280,320,358,396,431,463,495,474,507,540,572,604,636,668,700,732,764,796,825,340,374,221,168,110,157,145,200,180,236,281,321,359,397,432,464,443,476,509,541,573,605,637,669,701,733,765,797,826,375,409,224,118,115,102,146,202,183,240,286,325,363,400,434,412,445,478,510,542,574,606,638,670,702,734,766,798,827,410,408,169,121,116,106,151,134,184,242,288,327,365,401,377,414,447,479,511,543,575,607,639,671,703,735,767,799,828,441,373,172,122,117,108,154,137,188,247,292,331,367,342,379,416,448,480,512,544,576,608,640,672,704,736,768,800,829,472,338,123,81,73,109,97,138,190,249,294,332,307,344,382,418,450,482,514,546,578,610,642,674,706,738,770,801,830,503,303,126,84,77,114,100,142,196,254,297,270,309,347,386,421,453,485,517,549,581,613,645,677,709,740,771,802,831,534,266,86,85,79,68,101,144,199,257,229,272,312,351,389,424,456,488,520,552,584,616,648,679,710,741,772,803,832,565,225,89,51,80,71,105,150,207,177,231,275,316,354,392,427,459,491,523,555,587,618,649,680,711,742,773,804,833,596,173,57,55,46,72,107,153,131,179,234,279,319,357,395,430,462,494,526,557,588,619,650,681,712,743,774,805,834,627,127,60,56,49,76,113,94,133,182,239,285,324,362,399,433,465,496,527,558,589,620,651,682,713,744,775,806,835,658,90,35,30,50,78,65,96,136,187,246,291,330,366,402,435,466,497,528,559,590,621,652,683,714,745,776,807,836,689,61,38,33,54,43,67,99,141,195,253,296,333,368,403,436,467,498,529,560,591,622,653,684,715,746,777,808,837,720,39,19,34,27,45,70,104,149,206,259,298,334,369,404,437,468,499,530,561,592,623,654,685,716,747,778,809,838,751,23,22,16,29,48,75,112,160,213,260,299,335,370,405,438,469,500,531,562,593,624,655,686,717,748,779,810,839,782,12,9,18,32,53,83,120,166,218,261,300,336,371,406,439,470,501,532,563,594,625,656,687,718,749,780,811,840,813,5,11,21,37,59,88,125,171,223,264,302,337,372,407,440,471,502,533,564,595,626,657,688,719,750,781,812,841);(2,5,12,23,39,61,90,127,173,225,266,303,338,373,408,409,374,339,304,267,226,174,128,91,62,40,24,13,6)(4,9,19,35,57,86,123,169,221,262,263,170,87,36,10,11,22,38,60,89,126,172,224,265,301,222,124,58,20)(8,16,30,51,81,118,164,216,119,31,21,34,56,85,122,168,220,217,82,17,18,33,55,84,121,167,219,165,52)(15,27,46,73,110,158,159,28,29,49,77,115,161,212,47,32,50,79,116,162,214,74,37,54,80,117,163,215,111)(26,43,68,102,147,148,59,78,114,157,211,103,53,76,109,156,210,69,48,72,108,155,209,44,45,71,106,152,205)(42,65,97,139,193,88,113,154,208,194,83,107,151,204,140,75,105,146,203,98,70,101,145,201,66,67,100,143,198)(64,94,134,185,95,96,137,189,135,99,138,191,186,104,142,192,245,112,144,197,252,120,150,200,256,125,153,202,258)(93,131,180,237,160,196,250,181,136,184,243,171,207,255,284,149,190,248,132,133,183,241,166,199,251,238,141,188,244)(130,177,232,233,182,236,282,195,242,287,213,249,290,223,257,295,178,179,235,278,187,240,283,206,247,289,218,254,293)(176,229,273,264,297,329,261,294,328,260,292,326,259,288,323,253,286,322,246,281,315,239,280,274,234,277,230,231,276)(228,270,310,298,325,350,279,317,302,332,364,296,321,311,275,314,300,331,361,291,320,271,272,313,299,327,360,285,318)(269,307,345,324,355,336,365,385,316,352,334,359,308,309,348,330,356,337,367,398,319,353,335,363,346,312,349,333,358)(306,342,380,351,387,362,391,369,396,372,401,381,347,384,357,390,368,394,371,400,343,344,383,354,388,366,393,370,397)(341,377,378,379,415,382,417,386,419,389,420,392,422,395,423,399,425,402,426,403,428,404,429,405,431,406,432,407,434)(376,412,440,464,439,463,438,461,437,460,436,458,435,457,433,455,430,454,427,452,424,451,421,449,418,446,416,413,414)(411,443,469,492,466,487,459,483,450,444,445,470,493,467,489,462,484,453,477,447,471,495,468,490,465,486,456,481,448)(442,474,498,519,488,508,478,500,522,494,515,480,502,525,497,518,485,475,476,499,521,491,513,479,501,524,496,516,482)(473,505,527,547,511,531,553,520,506,507,528,548,512,532,554,523,539,509,529,550,514,533,556,526,545,510,530,551,517)(504,536,555,537,538,557,570,540,558,577,541,559,579,542,560,580,543,561,582,544,562,583,546,563,585,549,564,586,552)(535,567,581,594,615,576,592,612,574,590,609,572,588,568,569,584,595,617,578,593,614,575,591,611,573,589,601,571,587)(566,598,608,622,641,603,616,625,646,606,620,599,600,610,623,643,604,618,626,647,607,621,632,602,613,624,644,605,619)(597,629,638,650,657,678,637,649,656,676,636,648,655,675,635,645,654,673,634,642,653,663,633,640,652,630,631,639,651)(628,660,668,677,684,661,662,669,679,685,694,664,670,680,686,705,665,671,681,687,707,666,672,682,688,708,667,674,683)(659,691,698,703,711,716,692,693,699,704,712,717,725,695,700,706,713,718,737,696,701,709,714,719,739,697,702,710,715)(690,722,728,732,736,742,746,750,769,727,731,735,741,745,749,756,726,730,734,740,744,748,723,724,729,733,738,743,747)(721,753,758,761,764,767,771,774,777,780,754,755,759,762,765,768,772,775,778,781,787,757,760,763,766,770,773,776,779)(752,784,788,790,792,794,796,798,800,802,804,806,808,810,812,785,786,789,791,793,795,797,799,801,803,805,807,809,811)(783,814,815,816,817,818,819,820,821,822,823,824,825,826,827,828,829,830,831,832,833,834,835,836,837,838,839,840,841)Polycyclic presentation:
29^3|a^29=c,b^a=bc869 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(24389, 4);